Adapting dual wavelets to interval [0,1].

e assume that the conditions ( Biorthogonal scaling functions ) and ( Sufficiently fine scale 2 ) hold and build on results of the previous section. We assume that $N=\tilde{N}$ and .

We introduce the spaces and via the relationships MATH

The bases for and are constructed as follows. Let MATH The space has dimension . Hence, we construct a part of the basis for by taking and . The remaining functions from $W_{d}$ are derived by constructing the functions

(Wavelets on 01 step 1)

where

Similarly, on the right hand side of the interval $\left[ 0,1\right]$ we construct the same $\zeta_{d,k_{1}}$ and

by setting MATH

We form the linear combinations MATH

for some finite sequences

determined below. The wavelet dual bases of the space $W_{d}$ take the form MATH

The

are chosen to satisfy MATH

Let MATH for some matrixes $A,\tilde{A}$ and columns MATH Then MATH

We conclude as in the previous section: MATH MATH MATH

Remark

(Dimension mismatch) Note that we generally have MATH This means that we get at least 2 functions too many when constructing basis for the space $W_{d}$ . As a result, when we perform the biorthogonalization of boundary functions, at least two of the resulting functions (one for every boundary) vanish, see the proposition ( Linear independence from biorthogonality ). We discard such functions and end up with a basis of correct dimensionality.

Notation.

Index.

Contents.